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A Self-Consistent Reduced Model for Dusty Magnetorotationally Unstable Discs

The interaction between settling of dust grains and magnetorotational instability (MRI) turbulence in protoplanetary disks is analyzed. We use a reduced system of coupled ordinary differential equations to represent the interaction between the diffusion of grains and the inhibition of the MRI. The coupled equations are styled on a Landau equation for the turbulence and a Fokker-Planck equation for the diffusion. The turbulence-grain interaction is probably most relevant near the outer edge of the disk's quiescent, or "dead" zone. Settling is most pronounced near the midplane, where a high dust concentration can self-consistently suppress the MRI. Under certain conditions, however, grains can reach high altitudes, a result of some observational interest. Finally, we show that the equilibrium solutions are linearly stable.